Find F(x) given f(x) is:
(x+2)(x^2`+ 4x + 3)^5
Well my problem is what do I do with the brackets. I tried but I can't get the answer right.
My attempt:
u=(x^2 + 4x + 3)
u'=2x+4
u^5
u^6/6 + C
(x^2 + 4x + 3)^6/6 + C
I should have 12 instead of 6 on the bottom.
Answer should be:
(x^2 + 4x + 3)^6/12 + C
Thank you

You're forgetting the chain rule. Remember when you take take the derivative of that, you need to multiply it by the derivative of the inside. So, compensate for that when integrating by multiplying it by 1/(derivative of inside). The key here is noticing that the derivative of the inside (2x+4) is 2(x+2), notice the x+2 on the outside and you're all set. Feel free to ask me for any more help?

Since integration is just the opposite of differentiation, you need to do the opposite of the chain rule: instead of multiplying by the derivative of the inside, you need to divide by it. You can always check your answer by differentiating your answer and see if you get the question.

Part of your problem is that you don't work "cleanly". For example, you say "u'= 2x+4" but you don't say why you need to know that or what you are doing with it. The very next line is "u^5" which is meaningless. I suspect that if you had written out the entire integral you would have seen where you went wrong.

u= x^2+ 4x+ 3 so u'= du/dx= 2x+ 4 and du= (2x+4)dx. The whole point of taking the derivative there is to be able to decide how to replace "dx" in the integral. The crucial point is that du= (2x+4)dx but your integral has, in addition to (x^2+ 4x+ 3)^5, (x+2)dx, not (2x+4)dx.

There are two paths people use. From your du= (2x+4)dx, you can factor a "2" out and have du= 2(x+2)dx so du/2= (x+2)dx.